Even a 4096-bit floating point number can't represent the EXACT value of 0.1.

There is no such thing as EXACT anything in quantum mechanics (or basically, any science), so I don't see how that means anything? Every measurement has an error tolerance way above that implied by a 512-bit float number. You're way too obsessed with "ideal math" when it was proven over and over again that the Universe does not follow "ideal math" equations, but it is in fact quantized. (the randomness could be just a source of "calculation errors" too)

In fact, people have calculated digits of PI to an insane amount that we will never ever find in the "real world physics" due to quantum mechanics effects. Just because you can express something -- like PI -- with mathematical logic, does not mean the world uses such ideal representation.

"Even a 4096-bit floating point number can't represent the EXACT value of 0.1."

which states a simple fact in computer science.

I have no idea what you're trying to say here. I mean yes, that's true, but 0.1 does not exist in the real world. What is 0.1 to you? "One tenth of something" is a concept. Real numbers are also a concept -- precision is definitely not infinite. There's a point where you can't split something when measuring, so 1/10 of it makes no sense in actual data, since it's indivisible as far as you know. Stop mixing human concepts (which are idealizations) with measurements.

You can also say, PI is the exact value of "1 magic", with magic being a human-invented unit. This is perfectly adequate and you can use this unit for everything by converting it to others. It doesn't make PI any more "defined" than the infinite amount of precision it needs just because you changed units.

"Magic" is simply a number system using base Pi. We can define number bases as anything we like to suit our needs. Binary floats are just a convenience to us because we use binary computers. But it is by no means the only way it could be done. If we use base 10 floats suddenly 0.1 is easy to represent exactly. And as a bonus all fractions of powers of 2 and powers of 5 can be represented exactly. But none of this means anything outside of a mathematical construct.

"Magic" is simply a number system using base Pi. We can define number bases as anything we like to suit our needs. Binary floats are just a convenience to us because we use binary computers. But it is by no means the only way it could be done. If we use base 10 floats suddenly 0.1 is easy to represent exactly. And as a bonus all fractions of powers of 2 and powers of 5 can be represented exactly. But none of this means anything outside of a mathematical construct.

My point is that we have never measured something as being exactly 0.1 in size or whatever since it is literally impossible (due to quantum mechanics). Everytime we measure we get a different result because of the random movement and influence. Essentially, it is not possible to know that its size is exactly 0.1 (or any other), but only that it is +/- a small error around 0.1

Such error can easily be handled by a 512-bit float which was the main point. After all, even our own computer algorithms are based on ideal math, but the actual computations aren't infinitely precise. Same with the ideal math in the real world -- our actual measurements (which is our best knowledge of how the Universe works) aren't infinitely precise so we can't say that they are truly the same as our ideal math.

Just for reference (mostly to YONG ) 2^512 is on the order of 10^154 -- do you understand just how precise this is? have you ever seen anyone measure something with 154 digits of precision? Planck Length is "only" 10^-35 (yes I am aware not all 512 bits will be used for the significand, but whatever this is precision in full without even shifting with exponent)

It is one thing to change the subject of interest when the need arises. It is another when one just wants to shift the focus to something else in order to make room to defend oneself for his/her lousy statements and/or arguments.

Look at the facts.

The original focus was on simulation. So, we were talking about computers, or more precisely, supercomputers with the necessary processing power, speed, storage, and the like.

Thus, I pointed out the truth that the exact value of 0.1 cannot be represented by floating-point numbers no matter how many bits they have.

Then, Furs came along and argued -- by shifting the focus to something else -- that there was no such thing as EXACT anything in ... any science.

The keyword here is science, which is the study of nature by human. It is NOT nature itself.

So, I, once again, pointed out that lots of things in science were exact. Refer to my earlier post.

And then, revolution and Furs came along and blamed me for mixing up things.

In science, which is the study of nature by human, we construct lots of concepts, models, theories, and the like to make it easier for us to understand nature. That is just how science works.

At the end of the day, science is simply the human interpretation of nature. It is not nature itself.

Hopefully, a couple of forum members now understand their errors, withdraw their lousy statements/accusations, and apologize to the forum member who has been always right. Well, I guess that I am asking for too much, especially from revolution and Furs.

Your mention of 0.1 was completely irrelevant. It seems both myself and Furs were pointing that out. It has nothing to do with simulation or reality or any other topic that has been discussed so far. A concept that can't be represented by binary floating point values doesn't say anything about whether something is being simulated or not. Your request for any sort of apology is misplaced. Neither myself or Furs were attacking you. Instead we were correcting you. If you choose to view corrections as attacks then I am truly sorry for you. If you wish I could refrain from trying to correct you in the future.

So, the focus has been shifted again. It is no longer about "mixing up things"; it is about irrelevance.

See, this is how revolution conducts his/her arguments -- keeps shifting the focus to make room for himself/herself to defend his/her lousy arguments. Sigh!

Anyway.

Okay. Let's talk about irrelevance.

First, let me make one thing absolutely clear. I am NOT the one who brought up the "simulation" discussions; you are.

Second, I am NOT the one who brought up the discussions on the enormous requirements, in terms of bandwidth and storage, that a supercomputer would need in order to create such simulations; you are.

Third, I am NOT the one who brought up the discussions on floating-point numbers; Furs is. (He said, "I'm fairly certain a 512-bit floating point number has enough precision for the entire Universe.")

So, I pointed out, as an example, that even a 4096-bit floating-number could not represent the exact value of 0.1. Thus, it was pointless for us to claim that we could create an exact simulation of the observable universe, let alone the entire universe.

Please, by all means, explain why my mention of 0.1 was completely irrelevant.

Hopefully, you are not going to shift the focus to something else, again.

The bandwidth and storage requirements was in response to your claim that everything is recorded. I was pointing out that it wouldn't be possible to record everything because the requirements would be too high. All we could have is the current state, past states are lost and unavailable.

Your response about 0.1 not being representable has nothing to do with 512-bit floats and how that somehow means we can use that knowledge to discount any such simulation. You are correct about that statement of 0.1 can't be represented by any length binary floats but it leads to no conclusion from there. I asked you to show where 0.1 is in our current reality. And the point being that it doesn't exist in our reality so there is not even a need to represent it. Note that the form of 0.1, i.e. the ASCII characters 0, dot and 1, is not the same as 0.1 the functional value, or as Furs explained the concept and the measurement. I did go further to show that other forms of representation can be used to give an exact 0.1 if one wanted to, although they are not needed actually. And even so there is no reason to suspect that all simulations will necessarily use binary, or for that matter even digital. So the 0.1 value doesn't prove anything, it doesn't give rise to any new insight, thus it is irrelevant. In a mathematical proof style it means: it is neither necessary or sufficient to show that 0.1 can't be represented.

The point where something like a limit comes up in detecting a simulation from the inside would be looking for capped or truncated responses. That is, looking for places where at an extreme the value tops-out or overflows. If we use the example of 512-bit floats as the base unit for the simulation then we'd have to look for extreme points where an anomaly shows up where strange or unexpected values occur. Perhaps in a super massive black hole where adding mass no longer gives an increase in gravity because the base unit value has reached its maximum value. Another possibility is for it to wrap around to zero and the SMBH suddenly has no event horizon. Or a third possibility is to look for stepped responses that show precision loss as higher extremes are encountered. All of these things would be impossible to detect with our current technology if 512-bit is the base unit.

Wot? The response itself is not irrelevant, the content (0.1) was the irrelevant part. I think you are confusing form with function again. The misunderstanding about representations (the form) and the things they represent (the function) is the key thing to realise. I think you have misunderstood a fundamental point here.

Wot? The response itself is not irrelevant, the content (0.1) was the irrelevant part.

Ah, I see. The focus has been shifted again. Now, in order to escape the double standard charge, you need to split the content from the container. So, the message carried by the response is not the response itself. Very clever.

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